The {\em dynamic planar point location problem} is the task of maintaining a dynamic set $S$ of $n$ non-intersecting, except possibly at endpoints, line segments in the plane under the following operations: \begin{itemize} \item Locate($q$: point): Report the segment immediately above $q$, i.e., the first segment intersected by an upward vertical ray starting at $q$; \item Insert($s$: segment): Add segment $s$ to the collection $S$ of segments; \item Delete($s$: segment): Remove segment $s$ from the collection $S$ of segments. \end{itemize} We present a solution which requires space $O(n)$, has query and insertion time and deletion time. A query time was previously only known for monotone subdivisions and horizontal segments and required non-linear space

Baumgarten, Hanna

Jung, Hermann

Mehlhorn, Kurt

Baumgarten, H.

Jung, H.

Mehlhorn, K.

MPG.PuRe

Optimal Embedding of a Toroidal Mesh in a Path Michael S. Paterson Heiko Schröder Ondrej Sjkora lmrich Vrto MPI-I-91-126 November 1991 "Das diesem Bericht zugrunde liegende Vorhaben wurde mit Mitteln des Bundesministers für Forschung und Technologie (BetreuungakeDDZeichen ITR 9102) gefördert. Die Verantwor-tung für den Inh.alt dieser Veröffentlichung liegt beim Autor." OPTIMAL EMBEDDING OF A TOROIDAL MESH IN A PATH Michael S. Paterson Department of Computer Science, University of Warwick Coventry, CV 4 7 AL, Great Britain Heiko Schröder Department of Electrical Engineering and Computer Science University of Newcastle, NSW 2308, Australia Ondrej Sykora and Imrich Vrto' Max Planck Institute for Computer Science Im Stadtwald, D-W -6600 Saarbrücken, Germany Abstract We prove that the dilation of an m X n toroidal mesh in an m~vertex path equals 2min{m,n}, if m:f:. n and 2n - 1, if m = n. 1 Introduction The trend in architecturesof parallel computers indicates that two dimensional meshes with wrap arounds play an increasingly important role. They have the main advantage that they can be implemented using short wires only. Of major importance is their modularity, i.e. a mesh architecture can usually be produced of almost arbitrary size - allways assembled out of identical components which are connected in a regular pattern. A problem arises assoon as the whole architecture does not fit onto a single board. In fact commercially available massively parallel machines are implemented on stacks of printed circuit boards. Here it is essential that the wires connecting the boards are short otherwise delay is introduced or the size of drivers . has to be increased or both. This paper deals with the minimal maximal wire length of the embedding of a parallel computer with the toroidal architecture in a stack of printed circuit boards. The problem" can be mathematically formulated as labeling the vertices of a graph G by distinct vertices of a "host" graph H so that the maximum distance in H between adjacent vertices in Gis minimized. This minmax value is called bandwidth or dilation [1, 2, 9]. It is known that the dilation of Pm X Pn (mesh) in Pmn (path) equals min{ni,n} [3,5]. The dilation of Gm X P n (cylindrical mesh) in P mn is min {m, 2n} [4]. The dilation of Gn X Gn (toroidal mesh) . in Gn2(cycle) is n [8]. Recently, it has been shown that dilations of Gm X Pnand Pm X Pn in Pmn are min{ m, n }[7). In this paper we prove that the dilation of Gm X Gn in P mn is 2 min {m, n}, if m #- n and 2n - 1, if m = n. As a consequence we obtain that the dilation of Gm X Gn in Gmn is min{m, n}, which extends the result from [8]. *The work of the last two authors was supported by Alexander von Humboldt Foundation, Bonn, Germany tThe last two authors were on leave from Institute for Informatics, Slovak Academy of Sciences, Dlibravska. 9, 842 35 Bratislava, CSFR 2 Basic notation Let G = (VG,EG), H = (VB,EB) be two graphs. An embedding of G in His an injection <p: VG --+ VB. Let distB(Z, y) denote the distance of vertices z, y in H. An important measure of efficiency of the embedding <p is the so called dilation: diLp( G, H) = max {distB( <p( u), <p(v))}. (u,v)EEG The dilation of G in H is defined as dil(G,H) = min{diLp(G,H)}. q, Let X C VG . Denote 8(X) = {v E VG - X : there exists u E X, such that (u, v) E EG}. The set 8(X) is called the vertex boundary of X in G. Let Pie (CIe) denote a k-vertex path (cycle). Let VP• = Vc• = {1,2, ... ,k} and Ep• = {ij: li-jl = 1, i,j E Vp.},Ec. = Ep. U{lk}. Let Tmxn = Cm x Cn be a toroidal mesh. Let ri and C; denote the i-th row and the j-th column of Tmx", respeetively, for i = 0,1, ... , m-1 and j = 0,1, ... , n-1. Set H = Pie and for t = 1,2, ... , k, denote At = <p- l ( {1, 2, ... , t}) and Bt = VP• - At. 3 A lower bound For the lower bound we apply a method proposed by Harper [6]: Lemma 3.1 For any embedding <p : G --+ PIVGI it holds diLp(G, l=,vGI) 2:: maJC:t{8(At)}. Theorem 3.1 Let <p be an embeddingo/ the toroidal mesh Tmxn in the path Pmn • Then diLp(Tmxn,Pmn) > 2min{m,n}, i/m # n and diLp(Tnxn,Pn2) 2:: 2n -1, i/m = n. Proo/: Aeeording to Lemma 3.1 our aim is to find t which maximize the vertex boundary of At. It is easy to see that there exists t such that for each row rand each eolumn C it holds IrnBtl 2:: 2, Ic n Bt I 2:: 2 and some row ri or some eolumn Cle satisfies: Iri n Bt+ll = 1 or ICIe n Bt+ll = 1. W.l.o.g. assume that the seeond possibility oeeurs. Denote by r; the row eontaining the vertex Cle n Bt+l and by rz the row eontaining the vertex <p-l(t + 1) . As in all rows exeept r; and rz there is a vertex from At and two vertiees from BH 1! for eaeh 0 :5 i :5 m - 1, i # j,l it holds: Iri n 8(At )1 = Iri n 8(AH1 )1 > 2. Henee m-l m-l L h n 8(At ) I = L · h n 8(At+l)1 ~ 2m - 4. (1) i=O,i~;,Z i=O,i~;,Z Distinguish two eases regarding the eardinality of r; n At. 1. Let Ir; n Atl 2:: 1. This implies Ir; n 8(At) I 2:: 2 and Ir; n 8(At+l)1 2: 2. (a) H Irz n At l2:: 1 then Ir, n 8(At )l2:: 2. This ease together with (1) gives 18(At) I 2:: 2m. (b) H IrznAtl = 0 then Irzn8(AH1)1 2:: 2. This ease together with (1) gives 18(At+l) I 2:: 2m. 2. Let Ir; n Atl = O. (a) Assume Irz n 8(AH1 )1 == n - 1. Then all rows except r; eontain a vertex from At ud two vertiees from B t • In addition, r; contains at least one vertex belonging to 8(At ). This gives 18(At)1 2:: 2m - 1. If m > n or m == n then we have 18(At)1 2:: 2n or 18(At)1 2:: 2n - 1, respeetively. Suppose m < n. Denote by q the eolumn which eontains the vertex r, n Bt+l' All eolumns exeept Cle and Cq eontain a vertex from At+l and 2 vertices from Bt+l' The eolumns Cle and cq eontain at least one vertex from 8(At+l) each. This gives 18(AH1 )1 2:: 2(n - 2) + 2 2:: 2m. (b) Assume now that Irz n At+ll < n - 2. Consider successively the sets At+p, Bt+p, for p = 2,3,4, ... until one of the following three cases occurs: (a) H Ir; n At+pl = 1 then for each i, 0 ::::; i ::::; m - 1, it holds Iri n 8(At+p)I ~ 2, hence 18(At+p)I ~ 2m. (ß) Hthere exist a row r., s '# j, such that Ir. n At+pl = n - 1 then we continue as in the case 2( a). (;) H there exists a column Cu, u '# k, such that ICu n At+pl = m - 1 then all rows except rj contain two vertices {rom both At+p and Bt+p and, in addition, r; contains at least two vertices belonging to 8(At+p), which gives 18(At+p)I ~ 2m. 0 Theorem 3.1 allows to extend the result on dil(Tnxn, Cn 2) [8] for arbitrary toroidal meshes. Corollary 3.1 dil(Tmxn, Cmn) = min{m, n}. Proof: An analog of Harpers Lemma for embeddings in cycles asserts that for any embedding 4> : G -+ Clvol it holds cli4(G, C!Vol) > maxt{8(At)}/2. This and the proof of Theorem 3.1 implies the lower bound. Assume m ~ n. By placing consecutively the l-st, 2-nd, 3-rd, ... ,m-th row of Tmxn on Cmn we construct an embedding with dilation n. 4 Upper bounds Theorem 4.1 dil(Tmxn,Pt'nn) < 2min{m,n}. Proof: W.l.o.g. assume that m ~ n. By placing the l-st, m-th, 2-nd, (m - l)-st, 3-rd, ... row ofthe toroidal mesh Tmxn on the path Pt'nn we obtain' an embedding with dilation 2n. 0 Now, let us consider the following "spiral" embedding of the toroidal mesh Tnxn in the path Pn 2. It can be viewed as an n X n table containing numbers 1,2,3, ... ,n2• The maximum of differences of neighbouring numbers (consider the wrap-around neighbourhood too) is the dilation. A possible :filling of the table is shown in Figure 1 for n = 7 and 8. The dilations 13 and 15 appear e.g. between 23, 36 and 19, 34, respectively. The explicit function 4J for the "spiral" embedding can be described as follows. Let VT .. x.. = {(i,j) : -L(n -1)/2J ::::; i,j ::::; Ln/2J}, where (i,j) and (k, l) are neighbours iff li - kl + Ij - II = 1, or li - kl = n - 1 and j = l, or i = k and Ij - II = n - 1. Let [0,(0) x (0,00), (0, ~oo) X [0,(0), [0, -(0) x (0, -(0) and (0,00) x [0, -(0) denote the l-st, 2-nd, 3-rd and 4-th quadrant, respectively. 46 39 28 16 29 40 47 61 54 43 29 44 55 62 64 38 27 15 7 17 30 41 53 42 28 16 30 45 56 60 26 14 6 2 8 18 31 41 27 15 7 17 31 46 52 25 13 5 1 3 9 19 26 14 6 2 8 18 32 40 37 24 12 4 10 20 32 25 13 5 1 3 9 19 33 45 36 23 11 21 33 42 39 24 12 4 10 20 34 47 49 44 35 22 34 43 48 51 38 23 11 21 35 48 57 59 50 37 22 36 49 58 63 Figure 1: Embeddings of T7x7 and Tsxs · Set </>(0,0) = 1. 1. Let n be an odd number. (a) H Izi + lyl ~ (n - 1)/2 and (z, y) belongs to the i-th quadrant then </>(z, y) = 2(lzl + lyl? + Izl«2 - i)mod 4 + imod 2 - 2) + lyl«2 - i)mod 4+ +(i + 1)mod 2 - 2) + 1. (b) If Izi + IYI2: (n + 1)/2 and (z,y) belongs to the i-th quadrant then </>(z,y) = -2(lzl + lyl)2 + Izl(4n + 2 - (2 - i)mod 4 - (i + 1)mod 2)+ +lyl(4n + 2 - (2 - i)mod 4 - imod2) - n2+ n«2 - i)mod4 - 3/2) + 1/2. 2. Let n be an even number. (a) H Izi + lyl ~ n/2 then </> is defined by the same equation as in 1(a), except for the case Izi + lyl = n/2, z < 0, Y < 0, when the last term 1 is omitted. (b) Let Izi + lyl > n/2 + 1. Set </>(n/2,y) = n2 - (n + 2 - 2j)(n - 2j)/2, for y > o. Otherwise </> is defined by the same equation as in 1(b), except for the term 1/2 which is replaced by 1 (0) for the 1-st and 4-th (2-nd and 3-rd) quadrants. By a straightforward computation one can prove the following theorem: Theorem 4.2 Thefunction </> is a bijection and ifdistT .. x .. «z,y), (u,v)) = 1 then l<I>(z,y) - </>(u,v)1 ~ 2n -1. 0 5 Conclusion In this paper we find exact dilations of toroidal meshes in paths. For the practical problem of embedding a parallel computer with toroidal interconnection network of size m x n this has the consequence that the embeddings, described in the 4-th section are optimal achieving a . maximal wire length of board to board connections of length 2 min{ m, n} and in the case m = n even 2n - 1. In practice there is an alternative which leads to shorter off-board connections (but longer on-board connections), i.e. the toroidal array can firstly be embedded in a 2-dimensional array (with constant wire length). Then the 2-dimensional array can be embedded in the linear array using row major order. This way the maximal wire length for off-board wires is min{m,n}. The price for this solution is a strong increase in on-board wire area and thus in many applications might be rejected. 6 Acknowledgement This work was partially done while the last three authors were visiting Max Planck Institute for Computer Science in Saarbrücken, Germany. They thank Prof. Kurt Mehlhorn for his support. References [1] Chinn, P. Z., Chva.talova., J., Dewdney, A. K., Gibbs, N. E., Thebandwidth problem for graphs and matrices - a survey, Journal of Graph Theory, 6, 1982, 223-254. [2] Chung, F. R. K., Labelings of graphs, Chapter 7 in Selected Topics in Graph Theory, a, (eds. L.Beineke and R.Wilson), Academic Press, 151-168. [3] Chvatalova, J., Optimal labeling of a product of two paths, Discrete Mathematics, 11, 1975, 249-253. [4] Chvatalova, J., Dewdney, A. K., Gibbs, N. E., Korfhage, R. R., The bandwidth problemfor "graphs: a collection of re cent results. Res. Rep. No. 24, Department of Computer Science, UWO, London, Ontario, 1975. [5] FitzGerald, C. H., Optimal indexing ofthe vertices of graphs, Mathematics of Computa-tion, 28, 127, 1974, 825-831. [6] Harper, L. H., Optimal numberings and isoperimetrie problems on graphs, J. Combinatorial Theory, 1,.1966, 385-393. [7] Hromkovic, J., Müller, V., Sjkora, 0., Vrio, 1., On ~mbeddings in cycles, submitted for publication. [8] Ma, Y. W., Narahari, B., Optimal mappings among interconnection networks for perfor-mance evaluation. In: Proceedings of the 6-th International Conference on Distributed Computing Systems, 1986, 16-25. [9] Monien, B., Sudborough, H., Embedding one interconnection network in another, Com-puting Supplement., 7, 1990, 257-282. 

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